Problem 47 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ \frac{42}{x}=7 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=6\).

Step 1: Understanding the Problem

We need to solve the equation \(\frac{42}{x} = 7\) for the variable \(x\). This problem can be carried out using mental math, which means we shouldn't have to use a calculator or physically write out the problem to solve it.

Step 2: Isolate the Variable

The variable \(x\) is in the denominator of a fraction. To simplify the equation and make \(x\) easier to solve for, we can multiply both sides of the equation by \(x\). This way, the \(x\) on the left side cancels out, because \(x\cdot\frac{42}{x} = 42\). Our equation then becomes \(42 = 7x\).

Step 3: Solve for the Variable

Now we have a much simpler equation to solve and we can easily do this with mental math. To find \(x\), we simply divide both sides of the equation by 7. Doing so gives us \(x = 42 ÷ 7\). This simplifies to \(x = 6\).

Key Concepts

Isolate the Variable

Mastering the skill of isolating the variable is a key aspect of solving equations, especially when it comes to utilizing mental math. In simple terms, isolating the variable means rearranging the equation so that the variable you're solving for is on one side of the equation, and everything else is on the other side.

For example, in the provided equation \( \frac{42}{x} = 7 \), we want to find the value of \(x\). To do this, we need to 'free' \(x\) from being under the fraction. We can accomplish this by performing the same operation on both sides of the equation. In this case, we multiply both sides by \(x\) to cancel it out from the denominator, leading us to \(42 = 7x\). With the variable isolated, it's significantly easier to solve the equation using mental math.

Mental Math Techniques

While tackling equations, mental math techniques can speed up the process and enhance number sense. There are a few mental strategies that are useful when solving equations:

• Break down the problem: Simplify complex equations into smaller, more manageable parts.
• Look for patterns: Recognize common numerical relationships, such as the multiplication table facts.
• Use the 'invert and multiply' rule: When dividing by a fraction, invert the fraction and multiply instead.

Returning to our equation, once we have \(42 = 7x\), it's a straightforward pattern recognition step to see that dividing 42 by 7—which is a basic multiplication fact—gives us \(x = 6\). Note that doing the arithmetic in your head relies heavily on fluency with basic math facts and the ability to manipulate numbers confidently without pencil and paper.

Simplifying Equations

Simplifying equations is about reducing complexity to make equations more solvable by mental calculations. There are several steps to achieve this:

• Combine like terms: Merge terms that are similar to make calculations easier.
• Reduce fractions: Convert complex fractions to the simplest form.
• Eliminate unnecessary components: Get rid of parts of the equation that don't aid in finding the solution, such as zero terms or one coefficients.

In our exercise, simplification took place when we multiplied both sides of \( \frac{42}{x} = 7 \) by \(x\) to obtain \(42 = 7x\). At this stage, the equation was simplified to a single multiplication equation, which is easy to solve mentally. This underscores the importance of simplification in mental math; the simpler the equation, the easier it is to solve without the aid of calculators or extensive algebraic manipulations.